On the Ability of a Cable-driven Robot to Generate a Prescribed Set of Wrenches

This paper presents a new geometry-based method to determine if a cable-driven robot operating in a d-degree-of-freedom workspace (2 ≤ d ≤ 6) with n ≥ d cables can generate a given set of wrenches in a given pose, considering acceptable minimum and maximum tensions in the cables. To this end, the fundamental nature of the Available Wrench Set is studied. The latter concept, defined here, is closely related to similar sets introduced in [23, 4]. It is shown that the Available Wrench Set can be represented mathematically by a zonotope, a special class of convex polytopes. Using the properties of zonotopes, two methods to construct the Available Wrench Set are discussed. From the representation of the Available Wrench Set, computationallyefficient and non-iterative tests are presented to verify if this set includes the Task Wrench Set, the set of wrenches needed for a given task. INTRODUCTION AND PROBLEM DEFINITION A cable-driven robot, or simply cable robot, is a parallel robot whose actuated limbs are cables. The length of the cables can be adjusted in a coordinated manner to control the pose (position and orientation) and/or wrench (force and torque) at the moving platform. Pioneer applications of such mechanisms are the NIST Robocrane [1], the Falcon high-speed manipulator [15] and the Skycam [7]. The fact that cables can only exert efforts in one direction impacts the capability of the mechanism to generate wrenches at the platform. Previous work already presented methods to test if a set of wrenches – ranging from one to all possible wrenches – could be generated by a cable robot in a given pose, considering that cables work only in tension. Some of the proposed methods focus on fully constrained cable robots while others apply to unconstrained robots. In all cases, minimum and/or maximum cable tensions is considered. A complete section of this paper is dedicated to the comparison of the proposed approach with previous methods. A general geometric approach that addresses all possible cases without using an iterative algorithm is presented here. It will be shown that the results obtained with this approach are consistent with the ones previously presented in the literature [4, 5, 14, 17, 18, 22, 23, 24, 26]. This paper does not address the workspace of cable robots. The latter challenging problem was addressed in several papers over the recent years [10, 11, 12, 19, 25]. Before looking globally at the workspace, all proposed methods must go through the intermediate step of assessing the capability of a mechanism to generate a given set of wrenches. The approach proposed here is also compared with the intermediate steps of the papers on the workspace determination of cable robots. The task that a robot has to achieve implies that it will have to be able to generate a given set of wrenches in a given pose x. This Task Wrench Set, T , depends on the various applications of the considered robot, which can be for example to move a camera or other sensors [7, 6, 9, 3], manipulate payloads [15, 1] or simulate walking sensations to a user immersed in virtual reality [21], just to name a few. The Available Wrench Set, A, is the set of wrenches that the mechanism can generate. This set depends on the architecture of the robot, i.e., where the cables are attached on the platform and where the fixed winches are located. It also depends on the configuration pose as well as on the minimum and maximum acceptable tension in the cables. All the wrenches that are possibly needed to accomplish a task can 1 Copyright  2008 by ASME